Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian − ( − Δ)α/2 − q in Rd, for q ≥ 0, α ∈ (0,2). We obtain sharp estimates of the first eigenfunction φ1 of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim|x| → ∞ q(x) = ∞ and comparable on unit balls we obtain that φ1(x) is comparable to (|x| + 1) − d − α (q(x) + 1) − 1 and intrinsic ultracontractivity holds iff lim|x| → ∞ q(x)/log|x| = ∞. Proofs are based on uniform estimates of q-harmonic functions.